Vector space/K/Finite-dimensional/Normal endomorphism/Eigenvalues/Fact/Proof

Proof

Let

$\psi :=\phi -\lambda {\mathrm{Id}}_V;$

its kernel is the eigenspace of $\phi$ for the eigenvalue $\lambda$. The adjoint endomorphism of $\psi$ is, using fact,

$\hat{\psi }=\hat{\phi }-{\left(\lambda {\mathrm{Id}}_V\right)}^{\hat{}}=\hat{\phi }-\bar{\lambda }{\left({\mathrm{Id}}_V\right)}^{\hat{}}=\hat{\phi }-\bar{\lambda }{\mathrm{Id}}_V.$

Due to exercise, $\psi$ is also normal. Therefore, due to fact, we have

${\mathrm{Eig}}_{\lambda }\left(\phi \right)=\mathrm{kern}(\phi -\lambda {\mathrm{Id}}_V)=\mathrm{kern}(\hat{\phi }-\bar{\lambda }{\mathrm{Id}}_V)={\mathrm{Eig}}_{\bar{\lambda }}\left(\hat{\phi }\right).$

To fact